Monotonic use of space and computational complexity over abstract structures

We use relational pointer machines as a framework for generalized computational complexity. Non-reuse of memory space, dubbed monotonic computing, is proposed as a fundamental concept that threads together various abstract generalizations of PTime. Depending on the use of space, relational machines generalize DLogSpace, PTime, NPTime and PSpace, We show that alternating first order machines are equivalent, over all finite structures, to monotonic machines with positive queries, generalizing the Chandra-Kozen-Stockmeyer Theorem ASpaceif) = DTimeQf), and showing that the latter does not depend on any counting mechanism. We also show that of two generalizations of PTime, deterministic monotonic machines and nondeterministic monotonic positive machines, the former can simulate the latter on all structures, and the latter can simulate the former on enumerated structures. Finally, first order inductive definitions are shown to be equivalent to monotonic pointer machines with random selection. Research sponsored in part by the Defense Advanced Research Projects Agency (DOD) under Contract F3361587-C-1499 , ARPA Order No. 4976 (Amendment 20) and monitored by: Avionics Laboratory, Air Force Wright Aeronautical Laboratories, Aeronautical Systems Division (AFSC), Wright-Patterson AFB, OH 45433-6543. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Defense Advanced Research Projects Agency or of the U.S. Government. University JJhrariec